chapter 02: calculus with analytics geometry - mathcity.org · inu=xlni vj 801.1403; may‘ and...
TRANSCRIPT
9 ;».\ Duiwmilww 7;./\ ( c\v.\=.h,\ mo. 2 )6) 1
*?_~>"~~I?»r-~s? M f<*> 1» ‘Q ~»1»~w?‘*”“'x““" ‘MM M? -f0" clwld L.‘ f'(1_\ 4 Lb
xdlamlldh ,
f(~4\ = §—(fA+\¢\)--§?(1)
\'\--90 \,\
Sen; J;tM;1@;.J. _{,a-.J1.¢__;-5 w<@
A *\ _\® — *1-'Y\-\ 7
(9 3; (13 = 1
Q K-£.(_c,?) = ¢\£‘?.»>.&m‘A
_:_‘~(§“\ = O42“
at .-: -L)-L
“ 1
@@ _j;;_(q“+\,) = 'v\(m(+\¢3 - 0L
v <
@ %;(u1-v) =éa%‘;¢‘13’;
A - . \1.é£.@. .;.;(U.v) -u¢51;+ MQ ' U__UdV
3-(<73 = "___f'i¢‘?‘-'l"T*» _“I V7»
<1 - .41@_.£(cu\ _cd*_._ _.____._-V
,. I-.-.~_~_=-="=~— V 6“Mlm
‘-45.
_ __*______ - _____ 1
Exercise 2.2: Page 1/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 1/29 - Available at www.MathCity.org
A .5¢@CAX
<61) %;(c.m t_%<M~.yi
@ = $1151
' ® 2 -c05(LLX
é-— <$¢.Cl\ :. C¢u.’b,,,,\1 dx
I
;'® %(¢.~.¢q3 = _C',5u_1_Q§.A
c
|
; _i_1
r s¢@,16> ¢§;,,¢,3 _\___%
Ax J-C-£1
@ -3; = _:L_£!!--1
i ' -2 ‘
; \+1""
I. L ‘W = ._‘:® ch (Ci IS TE‘-L
i @ é_ Q;-4‘ = __.._‘\ Ag c 1> ‘F;
@ i(_c.=.;‘ ), =J1 L1 -;\F;|’
‘ su<®6) : (L&\,\~,L
@ -3-; = 9"-¢\'\z1
_4__ (Q.-Um‘ = -C..uz¢\\~L<14
@ (g,_L\,\~,§ = - S|.L\rw\-'§n.,.\A1k
@ =- -'0-'5'.-¢»\\1-%\r\1
Exercise 2.2: Page 2/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 2/29 - Available at www.MathCity.org
\L-L. ~-~ , _,_\.(D °h(S.w\\/\~A> jT___.
- SP1 @
<1 I -\ \.-on (C-AM ~A§ - --—-M_;
4 -\ \.;;( \‘aM\¢\ ~A§ .. _\_:E_
\
.\ _l___ii (Cik 1) - “llsL_ §Q<.¢\T‘~x) = -5-‘--‘A1 11 [1-1*
-\ __:_‘_______,_§:(CA1¢\¢\1) - *1-*-—'H_1L
_.._..--_.__ _ .__.__?._-._.-..__.,
$Zii\§—¢:v'1"*r;;--J-:.'» 5.» -,. M »»..;~;_..: €‘l$.E?4£LvIr - "'
’
Merging .*~*=.;*: ;§";=:¥~..w.-.-,.- ‘-,. ""r"‘>'7\'1\~»\y =.'..»,.'.,_'2-u-n~uu..¢..)-.>..-_';. . 1. .,..._ ,..,.; . ~».' r‘
+ 10' i
\'>\\L\
lv\\>!
(9 2
1
S
Exercise 2.2: Page 3/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 3/29 - Available at www.MathCity.org
EXERCISE, 2.2
Dixqérqntiate w;§r.t.(P1'oblcms 1-Ilii)
Sol. Lot y =‘\/WE'
o ‘J = (a’+.x’)"’
D34{.w-A.~$;;'1\.
l1~~(&+=¢> .;¢;;;~+\.=v».
I
,
9
%
Exercise 2.2: Page 4/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 4/29 - Available at www.MathCity.org
I2.,
@- -1 (a' + JP)’ "FA -1191-.dx 2-._. .l;-_.--' \
(of-v-x"\"'*' ~
. 0 ‘ “KN‘ I - V »';;;;;;&.i ~'»=*~
!3\\fa U-= J U + I M
~Jx‘+x+1 W\NV\],‘§C\
Lat y = (t'+'x+ 1)"'gig’. w..n-i-.-~\-v_|
ii- 1 {S-(1’*-n+\)y‘. §;(1+i*‘)‘ . -§H'n+|) ’. (La-\»\\.
2 -L"'—\_ *' ' I-‘§‘)'3 (x‘-uu\)"! t
v'2x+1_TT_1$“
G+I—
so
5?
_.._.-i-i.--_i--_-
T1
0+:
Let 'y=
+
7:151
-ac
51
¢I+I ~ -'N/~0.-+34,-'\j‘G—.7( .,_._
‘ '\', I», Qo\+v\~~u.\-1'.\'\l\v\\(\A1'|-»\.- A r,\.0-~v:wv~ \=1 5;;-_Ia_* _J?“,,\ Lia-atJ33} + _]'¢T.'{. L |a+1-Ft‘3-61;; (QPl§+\‘\-X)—l- ‘own SA—X
' uw- L111)‘ ‘“**> - \'*-*>= ‘Lou -2_’(a.+>\)La->\) La -1|“?-.~£14_
ma->\ -- A-\-1 '- Ix\j : “"'JE:7-L
*D;“,‘,,.w.,4. -X.x[0- -§_(a’- x2)" ‘(-2x) - [a-\/a"-xi]’-1
1E‘-1=
I.
¢
dx ' xx . ..—-——----a+ Ja‘-x‘
x J05-:5x
Y x‘ . x’+a'—x'v?‘_‘* °“""*"" “ff
= 'f ' 2 ="' 2 ‘X I
Exercise 2.2: Page 5/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 5/29 - Available at www.MathCity.org
G2
w/T{'““_‘\- ‘A xi '02-‘? ' \3&55091- L‘-k 3 I
I ox
80\
'22
T-F
Y a l;
I.
b’~\(- u-J\.t- x
'(sin_~H>= i , _
‘mi; cdlx( \/sin.1r.(c0s\/2'jams .'_ 2v;
‘\/I-;ain\/7:-(tin G)"COCK-MOI COS J1
_ IH (sin ‘/-Y)’ ‘
___‘/Y sin \/T coax-sinxcos \/ xF 2\/Tc‘\lIinx sin’\/Y
7
5.
§s:
J.;}Q-
oh ‘j-* *=/4..
" 1°K1o(-I2 + 1) _ ___._L.1. 7- ‘= ’17;§(»w> = |3-(\+\2 = p"*(*+\)
\ his l ll‘ -Mo ; M.
J.I"'- hM(14\)§;_“.g,.,|.J\-k"
\ \ IX2 --—-,..-----—-~ .
JI'n\v 1_§]w~(~IE+\) 55"‘,3, 4%‘. iL________.____...--- Nv~5'_y = liar! (sin I) 2.11:0:-.!]ml\‘-§\).(3\"-S!)
41 ““‘= '“""' <1-Y S01 dx_ = m’ (ainx) . J; (sinx)
2 ICC’ (Sill
I ($058.58
x). coax '
c’ (sinx).
‘1.*_ ,,_1§5'La‘a'
@
- 15' ‘T ~/T-~"" @<~-1:5»)= El 11
uh.
gz-|in\/;{;J.?W;- coax} -4\/gr§{cos\/‘:7--_"Ti_}1I
Exercise 2.2: Page 6/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 6/29 - Available at www.MathCity.org
G)
7. y=arctan(——'—'-L ) b 5’1-xcosa9;“. w..\..'t-¥-
‘iivs 1sotgli \ $_ 1.__9:--\ 5
1-xcosa)\
‘+ (wLS'~\c~]" (\--'1CI7>Q).L
(\-1€A¢.)"_ \ W . **~1->‘<<v4¢:~~**““?{"*°*?- z(\-\C»»)"+(1\$'-n)" (\""\c*>\)"
(\-\\€-.q)‘-
dx (xaina 2‘$1K\-XUW+
(\-1C.s*¢.3.'i;»o. -1\$~'-Q -(-C-M)
~ sina' ______ 5""‘°‘j _:_= I ' I . 1 . '-
1 — Zr cos a + x’ cos‘ a + x‘ __s1p‘ q__,_ \_;_.‘¢-4,, _\,~;{(|_-}¢,,,;,-j.)
Aaina3 1—2xcosa+:c'
. f+x+18' y 3 lnxz-x+ 1
Sol.__y u In (§éx?0- 1?h—tln (.1-"~x+ 1). LA). . . *.
.(.‘.2+x+1)_ (x2__,_.+ 1)
dx '(x’+x + 1)dx '_ -. Qx‘.-x+1)dx -
_.- .'>.1~.+ 1 2x-1 - ' ' _
'x’+x+ 1 '-x’—x‘+ 1*‘ -V
+ 1) (P-x+ 1)-(2x.—1)(x*+x+ 1)
'( (’+ +1)("- +1) '
(x‘+‘A+\\(aY' 1+- \=_1g?-{+1 -y;!Z_-1.;\_.'§‘-_--53-\ _)= --IQ’-u-2.
x2 I.9. , =1 l\~~\+nu‘-Al»\) <‘ jf ‘*7 j
O>\*1+\)(x.-1-M)
1. 'z_g!__!11(-Lt 3 = )2’ - (\
. \**+\X1"-x-M)- , -9~'\'\~ -also '--———-——————r———---M 1~3_L6'~'\
‘V01 = -in X?"
5
M. Kp.)~J\-",‘* ; VU_,,H-_,V,-‘,?.€1‘_\_v
.l_.é2. = 1‘.é_-+L~1-1*3 A-L 1 " m:2!@,,'%_!*J.!’?"i;{1,‘vi.i;~;‘}.l,.*;?f=T_}
Exercise 2.2: Page 7/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 7/29 - Available at www.MathCity.org
= x+2xlnx= I3 x (1+2 nx)
=9 =y.x (2lnx+1)
. =1-'2'. x(2Inx+'1)» =x‘“". (2lnx+1)10. ln(x2 + 1)‘S01 Let y = .1"(1? * *")
- - hDierentnatnpg w.r.t. x, we ave
Q1 _.__1 (12-+3‘)dx (I +x§dx
\ _ (2.1-H)..=
'__ '2.x+\-< ‘l.('\\-H3
""¥,,;-
11. y = (arcginxSol. Taking logarithmugf boih sides
-Qhj = -Q¥\($§:»:‘A)*
1,, = »Z'.*.:,.gs~.::=.;
BL“. w.J\-t- X
%1%41 ll; \ . A ‘~1> 0I I -.
*1
Now; letu =x"" \-l'nU .-.= .9.-~,1.’*'
M M {~41-1“
}_g_u_ 1 1 1udx\q;.»;,+iInx-
1 .
='-L + In x --xi xi
Exercise 2.2: Page 8/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 8/29 - Available at www.MathCity.org
a_J€,_,
1
--1.9% =-1'} ( 1 --'lnx')
_A_,_._,..~»:_:"'1-_,.,,J-A .» + 1%_ \.,v§~,; .1r * '- ' '9‘!'§
1 S=x"';5 (_1-lnx)
°'- = it-‘( \--Luz)
Putting in (1)
'3 \. ..
m; gm; 5‘~. ,,_,,.r:1
"7 »..A»= ”
iH”
vl’ IF’
1.4.2 *~ .;.______ L."‘ .ydxl. 1-w:1_X\___é+ %-~=~ 1 (v-L‘)V Q $2; . ~;.£,$___\__\_____+1“ .1“.-.),.s,.<\_,,}
$0.1. C\(wg~ .5 ‘it-_“
M, y=.'r'—9, iflxl as Am_§>\ 3£"A>>a
=.-(8-9), if_|x| <3 7* ‘* “°‘A 7D1141» 04- .1\;.1
32-. 11 =~¢\~I~\>/5
. ' =..-11* 4 W43
Sim}. ,l\-1‘
/'\
F;7*‘
Val’-
'\‘i \ ‘ -\1.-__-~ +1. ~».Lv\§iM‘A-1~'(\-1;‘l‘|.- 5 :\t-Q‘ V
L.-1
_-_
N
‘Ii----_-_._..
" \]._..
13. y= \/1+ N/x+~/'4;
‘Sol. Lu-. 3 - Ix-rD:-“- tn-./\.§:~‘L
ell 1 _‘____ ____:._ _ L (~14-Ivtv6' 1I‘A-\-J3,’-\-"IX; A‘ t
:.i1+ [1-\-ji‘
-= --M
AM-
!!l.9\i= Lxxt-.R
_2. i1.+\FQ.I-1" ‘W 7-I’
Exercise 2.2: Page 9/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 9/29 - Available at www.MathCity.org
I m ,
1*-Q xv:5°‘ 5" , if xjé ox‘ -
Maéz.
=..(,2r'-+1#;).”" ‘if M‘ o
= u=~>"* '~»»"»°]igF 0 3-L $1.0 ,'
Y»-Q. <.»~.,\‘;t\<'1, ' ‘
= lib-1)-2. = _.!,_-1-,. »J~; awe} {
T
15'_ Diu:'0n°¥§am' H =_ J;:*- ii LO
\)1+x'--,1] _x'2._ ,. " -2
Klarc n ,__...;i1
Fhta .‘.~.a(I\\-q,_\_+\..
.-pup,
w ft arc cos ‘x’r_...,x
*~' - -~ ‘.1-m.>~~=<-* *-~ ?
I-Gb U:C¢‘
ya
'¢='~-C-su. w=a~q,-.
\1.1+x’+\/1-x"
'= tan- A (
*-' + 0081]. (1§'c§)3(E' '
f_ ‘(.1 +.c<>s"1Z +.~/1-1-bad. ( _ _ .
\\ u-__‘ V 2cos" 5 - '\]v2 sun’ 5
= tan H L ,i " i .—_—"
I
§\f.\/_24co‘s2 g- + ’\/ 2 sin" g
tr ' Ti-Ca.“/,_ - 3t;;..5v(,_
U"C*°h. + 3'7-$12.."/‘_7
T11: (""u’l- ' SM“);* Qw;§§MU’L
' . 1 "‘*-~w~;1‘..,-9 \-'\@MU/t
k
'*T"‘“‘uI1.5 = -tail t“‘l 't,,__u/
SO ‘j = ‘K/H__U’L[ ~- L :'1~$ -= t::'*‘~(F/5"“/1)
NM’ ‘*>-A-4:-U ,
H9’ll
..\/L M
¢
Exercise 2.2: Page 10/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 10/29 - Available at www.MathCity.org
16. y = x'i“y '
S61. y- I xmwTaking logarithm ofboth ' V
cm 1-n
sldes, we haveac.“
ll’! y = -in 1 ‘J
L4»./\.-"\:-\\u
. A-Av\u.u.§1.5- _L..m ,g,JM='~“’/a1 = wag.“ L1 *5
P7. 1""-= ‘-x - 1
Sol. TakingAlogaritl'_ni\ -o§5tF sizes, we have‘N1? 1 -L“ ‘-’ I .Q
'5!“ = (1-~>).L.,.¢
v» .‘3L-m 1. 1-3
‘_'§-2-'l-P‘) :
. 3 +w;Lu.c.\5. v/4,.‘‘* "/a,-»qc.»;.9,~=¢-4“/¢, = gm 5
¢‘j;(x-u3c.»,3.L\§ ,_ §5g;_,,y,8% = ‘J$<-‘~09
" A1
‘L
ss (!m+\\ = 1
-'9
¢;.=<4*
5b.. 1A‘
@\~§~w\
3 -.-.X
\+1~\\\
54$
‘I.
\-Ix
-1'
K:4'
PK:
, ;,,._,\.§.y\_
\+Lu}.\ .. Xi(\¢\-LwA)""
\+l~.\-\(\+l~m)L
La '
(\+l~.\\)"
=C_
:C.
Exercise 2.2: Page 11/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 11/29 - Available at www.MathCity.org
I vjInu=xln
801.1403; May‘ and u=xYTa/ifingvkigarighmikof‘ both sides <3!‘ the rst equidtion
ys
Differentiating w.r.t. x, we have
In
ah2--—'-— lLl'Z+lnudx"x'ydx y
g
.7)Q
ydx-d=y' '3-E'E+Iny
-NV 8’-Pm‘!-3 -
alt luv =ylnxNow fr_om L _v __=q..r' , takingilogarithm, we get
Differentiating w.r.t. 'x, ' we obtain
l.@_Z Pix»v dx '1 +‘]nxd§
<£__ 1 <2dx ...v[x + Inx. dx]
=.x’[l + lnx.x _dx
The given equation is‘
ll-J-U=C '
Differentiating w.r.t. x, we get
Pntting the values of 5% andgé from-(1) and (2) into
we have
*3-"J1- V ' 'or (xy"'+Jc’lnx) 3*: ~\- [y'|lny+x". ]
»»is§:--~1
2232-0dx dac-
“‘4’/ +‘s‘1~‘~‘1 "'
(*§"'+>€’-1~,_>v‘-ff; = -nli~s-1»: +3-i.‘
+
>.‘~"i<
r—--1
-{RR
+.+
Inx.'. =
RR
f.l~'.~4.¢\~>;,;m =60
X ..
Exercise 2.2: Page 12/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 12/29 - Available at www.MathCity.org
. Q1 " Inor dx xy
w-.>...‘
1 Z+yx”-' ‘ "" +. y" In ' u '?~'§»:;r;,'§:i
Sol. Difforontiatirig w
" (x
.r.t. x, we have
£3) (1 +y') - (x +y) (1 _.y’) 1
(I-y)’ I " = 2r+2>:v’
°’~ 9"?)*‘*-12:?-<=~»>+§1m) um»U(-
I1.3_
Q‘\‘
3 -X-*5.) +(x-5)" _ 3
"H1-/§+xy§) = a(x+s;~;')(;__\,)"'13 * 3(“) = 2_(>\+3~;')(,_,)’~
..3+ I1“ = (""'5'>')(1-5)‘
ma-""‘~
(u-5) = x(;_-,) +3‘*-‘Jkx-a)°‘) = §g+x(x_5)?-
+ Q=J_1=LL>Tx*)'(x-y)"
'20. x+arcSiny ='xySol. x+ air?-y =vxy
Dif'Ferentiating(1) w.r.t. x, we have
".1+
.-
@ ,
O -1
$3"
O'1
Q-JG.
1-="~<
r
U\_____J
Therefore;
I-1;]
2~F:'“
3+‘?
1._
' *d= xa-‘E4-y.\
—§L%; = ,5-\-:1---——__ = -
-
/Q--\L<‘,.*<
NIi‘VaL\/
~'~<
'D-4
._x 1- .
1-)‘=y-1,
ix ;'£x_____L—1>‘/1-2dx _ 1-x‘\/1—y2 ‘
\ .. __ _,_
Q5!‘ . - E
5 1 F
x i
t
Exercise 2.2: Page 13/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 13/29 - Available at www.MathCity.org
I|
»
In Problems 21-30, nd}: (;):III. f(1) = In (:41/x‘-1)
SOIl- “Q1 1 -Q-v\Kx.y.‘.*m_\)
\$~\&~-w---i-1Lf(‘l§ = .
‘ Qulgrq)‘ (W =:J'~I*'.Tu)
= (\+)(1\+[F:\) J1?-\
= -—‘-— -[_E"'§.~..:1.](‘»\;*.F\T-_n) 55"‘
22 I:f~f(x) I In Te,
801- fu>=1n
_= In e“-In (1+e‘)-H1) = x-In (1+e")bZQ.w-»A~‘Q-)4. '
‘ f,’<x)=1--—"-—-c*=I--5,-L-—1+e' \-\-Q‘__ 1+¢'~e" 1='*""*"*.i—— -it-'1+‘e" '1+¢‘
23_ ;_:.-£;.xIn.l-’ V.
. | I '
Sol. Taking In‘ ofboth sides, we get
In 0 <1» - In 3“M-on fl!) 2. -9»!-Ln}. '
m..A. ‘\.-1W.§(1) ‘ Luu.-%+1~\\-I,-L
fm =§(~nI'='m_£..*1'0-\\ ‘
= 1 I’i::-1Jun-\
= X '
I
Exercise 2.2: Page 14/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 14/29 - Available at www.MathCity.org
H=*\9~ gm = ‘sq 1:,';;§~'.,;;I:...1£)- &“(_1t;;.-3T.1+M.§3}U4 . u.J\. . 1.
24. f(r>==1" -———-“H "1-VTV
Is
I
\S3
4
1
i
S01. /<1) . In (1+~/T)_1n (1-~13?)i
.
I \__ __L.V5(1) = tfi - (O
.. _L,.4__ + 1~‘ 1ym».y~A) =-(\—5\>
s J-—- '\'1? \-\-ii \-_F\_
= .1/[\.;£i:_\:=.Ji~_. K
1)“ <~~r»\>u-rm_, ,\___.__.__1 M»lji (\-x)7-
_]§\(\-1)
35- f(;;) = e" cos (b arc tan 1)
3ql,'f(,¢) = ¢" cos ( bt§1‘\ x)\>;4§- w-A-t-1 Q‘
;'m = 2‘. -sc.,.(\>h:2,q.b.1..= )+c»(\»*-Il*>- e. -w\ ML"
usé
= ¢ [_¢.c.~m.::1)_- %;,%=»~\\~*-51*-F]
;~}V»(\~¢) .2Q,§5,;1\ ..\>s;,..&.\»*»Ii»\-) X<@;§GT”'1‘ L % %
~ \1b+a+\/b-atan‘ i28. f(x * <1
‘ b"a ‘/.5_:z—\/_5—;a§an(-)
[ORNR
Sol.
.._[\jf;.g,,<_2§__.L. _\ \ ,_1'».'.".~Z1»-~--———-———-—1=f“) ‘ m={ J'$K+JTT;.m..-:1 ' ‘ ‘ T‘37~'F7—1"'~“!L
s
|
Exercise 2.2: Page 15/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 15/29 - Available at www.MathCity.org
{ho = _~_ 31-lb-~~$~1=:‘
~r
T51
M7 1 ~~ + ' . ~
lb-Q? ‘B-HA +5: B-Q-Q - B Q mi/1}1
P‘*111'
PF
:1
"“'_1r“-""1
0’
._ -- \_ _______ .____-_ ...___ _ ._____.._____L__ _...4. __:.‘zju»-sum) J -\-A '*q~,),-K tr.»-g Jxm. -]\»-~hw1g-
‘I. 1 _______ , '= _:§i_:_.' . lb-\-Q -‘LF-A t§’~j1!~L +- i\>—\-Q i9LQ'I-L24%/Q 4%-m (,|\».\ A-{KI hm:/1)(“';'“ RE; ‘ahv
= _$i1-‘_I§_:_ n7_,d¥>+A..1ll»/» (‘>4-Ck)-Utv-A).g‘_:_T§':_
SJ 2* CL» it-_
= 7 K x W i .1 N =(\u+\\(‘,, L -(B-Q §~M 15-_
c.,>‘L1.
1-‘ 'l-1_-.C -Q1.‘ J> ;' \>c${-*+<\c.‘3§ -\>¢.-}12_ +<\%»1;1,
\
_ m (C.§1;+¢.-£}§+bLC~§-‘§—_-Sig)
\
Q a. +\=Cus~L
»27' f(-U = §ra'.5inhx84" ff" x.¢:.SJ.~\vm
.u..,\, .3;(M = x.q‘5.c.\\“ +1s.;m.8.‘-J~.¢ + I-<1-$~l~\w
4 ,, aL‘[x€.;\\1 .»xS&.L~1.l-¢ -\-Si~\w\;cos x 1 x~]f. f(x)=---fa?-;+§lntan§
Sal» Hw {(1) = ._.L. c¢*__ +J..,Q~\bM§__- . 7- SL3: 7" '“
L.)-v‘--'2.-\\ 1f(x) - -L $Q\u.L.€-;~‘*\-CJ>\\.1$r~'-\\("**_‘8+.!_ . -L-— -
1. 17> L “M3; 2. 2
1 I__\,. .
1 $,,;~‘1“>\‘ \‘ *6-~!.:
w
I‘
gvi
1
1' - sin’x - 2 cos’ x 1 co's§ x/2" 2 sin" x + 4 sin x/2 '
’ ' cos x/2 ‘
. K5
l
Exercise 2.2: Page 16/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 16/29 - Available at www.MathCity.org
-sd;1—1.&'§:_. .___\»——-v"_“""_'_'% %
"‘i{"""'v=M»L 1* <~=>>»%<~»*1> "»
3"‘ . -.smsx_1_ - 'si_n’;x--~2 cos‘ x_ 1
2]+
2.(1sin is cos
sin’x+2cos’x 12 K . * . -
~ -2 sm’ x Y 2 sm x -
_sin"xi+ 2 cos” .xf_+ sin’; ”_7:‘i§>:l~5-xf-i7(__Cg1§___"§-_
5 2ain“x - ' 7.9521
j 2$s"»>=\+<;M _ M- = Q2“I 1%; ‘pi29. f(x) = arcsec (cscx +_\[; )
54“ Wu gm, ml‘ (Gu.u1+_\'T')$44. w.)\.-'\'.-'1.
é('\\ = __.‘._- ~~ ‘ ‘--;,"_~_-__*_'p”_':;;";T_ZT'_"_'I." (clli 4»-Ii) .
' (C~u<1-vl-\&')\'i (Gmm +33)! -— \ at
= _ - 1 - (-C-v.u\.€»\1+(¢,y_¢__1-+\/;)'\}(<-u.cx+\l;)2-1 Ly
4- z_F\C»u (Jun -3-J\
3 [CAI-ml -\-.5) Tau}: +1 + zfl C-H1-‘l~—lK' ly‘1-2\/;cscxcotx(ml} + J; ) \]¢-»>e"e.r + x + 2\/Zipmx-\ ~
_3()_ = 1+ ;1 Z .
Sol. Tgkinl ‘“'°“’°““=id:§» We have.
.4‘-y\§(_1)-; RM
‘I . _
us" wuhkni 1 \-£1-\-\).\ an
(\.‘» .---~
- ~__. 1» ——..1.._..§(1\\ -.: §€'._:_\..;;_. _\* (x3 U‘
'5 1_L_‘ {Q0 = 1::;!~1 4-'Lx.§-
ft!) *"“§\‘;’fA;__\ kkxjl) A 55;.-
1 (V1 + \ ‘» x. wwvv.:~":mi $1
Exercise 2.2: Page 17/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 17/29 - Available at www.MathCity.org
‘*9 = §(1)“1xQ_,_(‘+'\___ ___\
_-ql 3l(3l-M)
‘Q
‘= (\*'3;)[7-"*-Qmlx-la) _. __L__. ]L “_' 10m)
Di‘aren§ia'te‘with relpgct to x" each 30!‘ U'>c"f'01]0Wi'ng
(Problems 31-42) I
81. arctan 2-xSohld’
_ _.| + L3 * *~~ ( T3‘)
lm11‘: \
. (_g.J\. -k. >1-
4 \-H1L Q;
= _ ‘ (1-_:‘_7-7--(\+u3»(--\)L ~..-‘_.._A_ .
->‘1-11 #\-b-11= \ -~ __._.____._\
..__<1-»>‘~. _@_w.>%% <=~->*(1-1‘; '-
(1/-1)‘ s s 5 \: — _ ....._..-_ __‘_ - v » ‘. _- - - __,,__,--. WW - _-—__.» _. _r -‘4-lm-m‘+\ +‘** Hi (1/J-a_\"— 5+§>\_‘ S(\-\-1"\ \-\-->2?
132. ln(arcsin e‘) + yx‘ =
\ I 3
S°1' ‘"'-~- 1~\(uI.£‘) +31." -. \h;..A.‘\'.~%\.
-_~-_~_.¢_ A. 1» =‘ d‘(‘F-.e._ + A‘ (31) O5=7~:(€‘_
‘L--—t———- -________‘ .€?‘ +§\-?.1\-¥\\.éj-~-:05-ll H?“ \\_ (£151 *
\ ex 13-~_"~-—"- _---, +. 2.1 +1» 3-_ = c9<..l(e_*> 1"'_'\_em 3 4,‘
1 A lX. ..‘L. C
. A‘ ' 1-‘Pl - —:‘—-— -- _‘ V Q-(e7'§ |\-6‘ '
@ &~ : -L __7-*\ _e__,...___‘ -~ 15' C.“Q xi, " ‘J -. I-4 ‘ = _ iv —I 'i"""""*"_"'_"_"'_':'I:-‘ ‘ - Q-~ G. F“ 1, 1 .-' 1 \
‘» > * new L0. Mg
1 IQ’
Exercise 2.2: Page 18/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 18/29 - Available at www.MathCity.org
.
33_ y .-=i (arcsin 3:2)?‘.-| 1‘
Sol. Let 3 = (""1 *2)Q2“. w-A-4» ~A
%,. . 1<[$}:\‘s\‘-_\‘-’- ¢;\<==:-*>‘R
..\ K "‘= 1< (1-=.. M . -‘----_-.-.~. . L1
X \ (‘\.\\
5"~15
‘L11: \$I~3~‘ *f_')f_—‘_
_;\-uh
a4. arctan (1-')+y1“ = 1
\7|
?,£
~51‘
N1
. ,
S01. G'\\u.~»\ '\‘¢Zl(5Iv.\ -\-'3: ‘ \
'>3»\£- ""‘~'*""\ 1 Au -‘ k1&9}¢‘:!3:_\_"§ _‘_ 3.1.1 -\-"A. Tu .. r
\+\”I\\\t ‘P \A,, Q
\ ~A'\"kw‘°_A\ 4. Zi-\"*1"‘ "
' "“" ‘*% =°\ ‘*‘§""’\ + '*-*'~*'** "‘
£25; L‘ {AW = 0* “*1. -\-‘1~*‘.\* '3-I
-%t"§," "’¢!_:z_
‘~35, J’ *~g‘é:L ‘-113 L \.\"L__.§:1-;;- ,3; ‘J _’~__'}(‘ .,\ =-1.»\\bu»1.
1 ‘hi \ ‘ " ' kt'% ~ 1 1.\‘ CO5 X M \
35" 7 = 1 + cosh x \ "Ida (_*""* ‘*9 ' -.3; U5 \-‘L!\(‘A"') ,7/P§v~§t
Sol. \" “’_"*:~__f__-:~.C! G .
c;.;. .1 \--r.»,»\l‘\-M-A\~»\
9;-.A» ‘\.- K9%; . (\~c.Jm\.%;(\.f-‘A13 -. (\- <'-»&u\. (\4-W~*~)
<»~b;~\.;S’¥'"__'“ % W“q+r.,.m)g-$_..__1,,- \\§_g\.-<:.,,$§_1_\,\;.<'.'-.1.~\
(\+t.An\\‘"2 _‘:_&_1!\ll_:_§':_§‘:\__j1£_$_',J,~_§\ +G‘.\\-‘J-R‘ "2'g'\A* ._._
» <.+¢.w '** ‘ <~+¢-Mx‘, -1-z$g,§~‘\;§__.1-"lb. .-l4Q,;,,_M1;.LC-J»‘4!~L $¢..\\*h- _\,,... =..\3..\\!_uJ\_l"""' 1-. . ' - I Q q----"' ‘ ‘ '_
L=*‘-*~“‘*/1.)‘ uclm» C-\»."h. ¢~%~“/1 ‘- ‘
Exercise 2.2: Page 19/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 19/29 - Available at www.MathCity.org
M1- 3 '35- y = In (¢lnh'2x) '
"0"" ('5-~\\2.1)
B_"“- U0-.A."\L- wk
.-_ \ 4.,1: **--~‘ 3'; (+A~»\\u.)
-hank2.1
1\‘ -———— . $4~¢\\11.7_"K-u\\ 11
‘ c‘u\h\ - ‘ind-3‘'11. 1,
. G1/‘\1.\( \_..._._‘._ _ ________ 2__
§'\-~\\1»‘£ cs!5\:~~|_1
1 u‘ '§T'EZ§ ‘ '""**~"'~"" W
’ ‘A 1*’ vk 7~°~§~-~~\\u\4-C;g\\1,1_
S01. ' y = ]Qg‘w'
g_>’“"\
H
$3"
ll
“.57F
'5'.8
"#3-i
4* J-m,' 1+» "X: _.._.§_._.___..
(‘+\>~LW\\7. QYH) ' 'a“;T\]'_;;O_x_1_
>-1I-Z\._/
= -= '-1¢"~u.<.\\'q\
l x + 1 - 9““\“‘* --sv. l0g1o(T) r ,.
1“ (‘A4-\__._:'=§l
J“/\ \0
‘-1
- (1-».>.>]xi
c»x --\ -1
_..\"" ____
38. arccoa ‘J 1 -xi _ (.4,r"'3-]‘1,_\,_, .3; .
S01. Let y I _¢$§\)1__,,W’
.é:2A _ -\A - —---- i=‘ ‘-—--—\_(“Y_‘_:_,M, Aihn >0)
= ——-—-- - -‘-»-t-11> _ =1
-s.{W;=T 1-F31 "
Exercise 2.2: Page 20/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 20/29 - Available at www.MathCity.org
A.-
. 2 v ¢ ,A 4» ‘ -15‘-4 *""
1; ' - " " -;~ >_"'~ 0.717?§i“i{?“!'_'.‘4%%Z"g‘i.."§i‘.~ Q51
VJV\f‘N.if'i~§Fz :;;}[email protected]&i},",&Il=%’§‘
\ .' 7-I
'r?5'W‘= *_._‘__._
3‘!-
\><s F-I1
cum 92¢ ($1;/v\\'\‘)
$3" I-u< j= <.¢z‘(s¢“\A>\)bL1/1],. Q)./~..E-‘L
4»ch
- \ ‘ A §%iM\A_x §*'*$~.~;\\“A1*\l§4.~/\\v\‘A-\
-_- __L____..__._< \ CJ»\n\
s'i"'\\'“( l&g»§»\/\\I\L"l-"\
CJ<\~~_A____
X?“-\K0. c>\Ac$L~\(<uL¢c,9<,QM»\)
M-
$»9\. Luk %‘_/;.\(C£‘JQMM_§
»'~\s- W-~*~"‘* -all , __\ ___;_:.-. . i-(c.;x,‘$lM§4* §\-§c.£‘nM»\" A‘
\ _\ . ,\ 4
’ 1\-\-Ll-ML 7‘
._ "\ ;"""x(_\+9l.1§ J \ -" (C-i"~Q-*\ __’/____‘_/,.
C.a\(‘(u-1‘)
s.&.- Lu $3 = ¢~!~\/?‘§\**“)
A;
g,q.J\.¥:"k
__ \ __ .11‘-(_\+‘k‘"\
Lil-S-91-
AA»~X?\‘;’_tI3f:\"3K1* 1* _... :I--;. ' :: -—--'—'_”"' ‘L.____,__W _), V“ F‘: ~
\
L-2.} ‘j%?§.‘u.A.£.f ___¢;£_5;\Q6g__|__'
[R L +...\§~\ A1 , f§I¥5§\§"v.
a
K
Exercise 2.2: Page 21/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 21/29 - Available at www.MathCity.org
‘I-L
Differentiate, (logarithmically) with respect to x(Problems 43-47)
I-—-¢——-—_.-1-. I 2
I
43- Y = “*1
sols-La 5\(1"+ 3 "33 I
3 ll (1-0*]bu.‘ "'N\ IV\. an Mu‘
J/“*5 = 1“ L *(\'*4\_)__}V'i(1-1)‘ '
\\(1‘+|_§‘ 1
- (vs-1)‘,' 'l5'l’-l-at +.l~.(\\'-+\\_.X~.-.(_\-.\f‘.s
‘*3 " J5 l_-‘l’“‘\ +-l-(1‘-H.\ ~11“ (1-0']\1\%- \'°'~'\-‘E-K
- -\----2.1 1 l' '-’—*"\ 1- L.-LL--_7;__l ,3“ 1" 3l ‘* wk‘-a-\ ‘A-\
wl'
¥°
W
wl’
3 d‘
\(\‘+\3(1-\) ll ll ’ ‘_.__\‘*..l‘_T'*l'}_"‘,“}~",'7_"‘§,§.l'rL‘¥-l
' J=-l.;{(\‘+\§(1-\\+1~A-x(_1_\)-'z1(‘5H)] \ [1 2 1 1- 3, _3
3 \(~1‘+\\l7n_,).l;.‘é$?_-k = M1
- ‘*(\\"-\-\)(\\-\) ,l
An '1 - ‘I ‘I'% _,_.. _A, ' X-E£__~"‘~.‘ =
13-(x‘+\)1' 1”’-M33‘--=\~\'\.,‘ 7‘*0 v ~14- "'“””""'“' “"“" '+ >( U 3(_X-\}' Y (X-H H1-\l'3 1.
= L’-§:;:1_'l __ __ x.3- 31'.‘-1,-_\|_\,1 §7\§‘ |_\I - .__...__~_.._.____.l,._. _.-----V-M -- - I5.-\.\3* '(“"7 '(‘L*‘) 3 3:15. (1--\\.q3-‘(x‘+1lh44, y_ \r§(1-ix)"
_(2"';_1)" (3-4x)'"S°]' Tam"! l.08arithm of‘ both sides we get
"“'\‘j : .1/v, [I*T('-11;“. _(**"$""""'“"v|1-an ". (3~ln\) ‘l
= 1,, "; 2,,’ '1);<5 (\-11) l... 1“ (V1 gx) _(.5_\m>
= (jx +l“("“?l ' ll“ (’-~3\;j"+ L (a- w ll/El l= -R11 + l~.(\-1\'\A)’3' -1\»\(2_3“)"~_ 1M(_5_‘w)"I3 ‘V
‘M3 = -%~j~~X -\- (\-Ml ) - (z_},\) (3_q‘1)
Exercise 2.2: Page 22/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 22/29 - Available at www.MathCity.org
bLgg~ U-A-{V-K
: .\_. \ 7-3 6‘ ,~-;_- +-3--T‘-I-4-1)- .1‘.-l_-kn -1- l--(-*0
— 1 H Z pl. '3 3-‘I1
-3 _ *.\1 ‘.\(\-11) \<(z-1,~;) 3 (3_“ i)
2 -3~—1+ " A + Z“ '7 M
** "<1-»1> =<~-1» * '-;rT"1- 1)
= 1“-3‘3,*§} 4_ _-‘4('5--‘<\\+\'-(\-u\\M‘ ) ’>(\_§_‘) ‘)7
, “""\*"* + -n +\£1 +\L-31%
‘“‘ U4‘) 's(\-1.» )0, -R1)
-23- : ‘H-35K H_\L,\(
“‘*‘\(1-'51) ‘5(\-11) ('3 -K1)
- '& ‘A i +3‘ _ + t\—-\L\J1 Riki‘-$1) '5(\__1\)\3_\n‘)1
: _'Ei__(_~""'x) I’ * ----‘K + \\(\_\\)____
(14; 3”" ('5-M1)" “* (7--3*) 3(\-.u\)(3-R X)
Q5» y = (tanx°°"' + (cot. x)“'"" G) '
Sol. Let ui= (tan x)“""‘ _
mg 2“ PW 1.». #1»lml.) =In u =‘<;0t'x In (tan"x)
DiITero;I_1tiating, w.r.t. x, we have
1_.é1 . c.»\=\._;_..s¢t~H-L»('\a....3.-c.ué1\U 3' *3.-1
iii; = U {C4\\.(-.**_ué‘ __ L-(‘\'n~.\\§.O\|,;"\\=(*--3‘Kc,i‘,.~;.h - 1-(\'-~)-C~u¢§\‘S
' 1(+..8‘[' :6; - 1..(v,.,;,¢,.,},
*5; ‘<*~§3‘[¢..,¢* -1_(x.m-uh}
A \} ':
12“, = L.(c.¢n **‘*“_= \:~.*.1..<c¢1)
‘*1-u-*--k-v.a 'b...~g...L,..-G4-é‘1 +1'*((:*1\,S¢l
elm (‘J1ii‘: = V‘_—‘\'l-M-'\'4.~i-G-1:! -\-1~(C~u\_.$lr}*\
= v f__1.1.1.c-.48» + -°~(_c.¢,\.s¢b\\
. (Q,:~[_ ~1..Lc.m. QZJ]
,/J/A z (_(,§,3 £- $1511 -+ L (U<\\- <‘»u\u‘§
Exercise 2.2: Page 23/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 23/29 - Available at www.MathCity.org
New ‘Q7,
3=U+\J1*‘!Q1“ - 1£1. .,'
d1 *,
Pmtq vJ-u>A» QYTJ ’ (*““‘*3 ‘[Q“2“'£‘“(*"~*>"c‘-“é“i + ((J=‘:;M‘{"‘iI¢}\ Q-~9~~(r‘.h).%u'I1L
46_ y=x‘.-e‘ sinx-.lnx '. “MS91. Taking logarithm of‘ both sides, we get
in = 1» (13.8-s;..,.Q_.)
g J'““?*!*'*¢‘+1-$L.1+-l-(-1-'il)s3\-Ln! Jr‘!-~Q +1mS.'».'a-Q-L.(_l~,\A)
-3»; 1 ‘A-Lu +1 -1» .1-'i;..~. .;.1.,\(.lM\,)$4,‘. hi--A-k-K
' x"\i"'L*"l+1 + -L--~-C-M 4. .L_. .\_.<‘-.-.‘ 1,,‘ 1
. 5[l+l-\+ i +c,&,\ 4, _i___
L‘ i . “*1= 1.¢.&..1.1~.,)[1 Q+ Mi +C'Jn\+ £71-ii} A_____..9M\
Q.-
i“l<‘2~
In Pmbleins 48 - so, n
(1 + 2)’47- y=""“""'-*1:-——';r
‘.(x+1_)(x +3)
Sol. Taking In of both sides we have;
. z 3_ 154 , (‘A4r,\3V\’\-\-\_)(\1'_* vi-‘_*lL\_ \%1L J!‘
48' ‘G + '\{_- _ ‘J-5 *3)Sol. Differentiating, both" sides w.r.t. x , we get. '
L - 3 'iv)‘ &“[ j ' 'q"‘(“’“l"'Q““(“*\)~»°-~(\‘-n\1-is - 11-_u+_1)-l_41+\)-z.L.(_»3‘+ 3)M. \o.\-v\.§. '1‘-~- . 1..L_ _ _.1_» _3..L_“-_-21
.5 4‘ an 1+1 £4; x 1Q; - ~,{ ._ .L. __ ix f zumu‘-;) -(1+g)(\m-¢\(».;;,,,)+1 ‘Nu ‘us * '1 ( ‘, ‘ ‘ L \\+\)bu-1.) 1+1.) J- '1 Pl‘ “"“”*‘:\ *<'3:('* -21:‘;-1_§;1f_§*(1‘+s1+1_>‘AM; _\\-gm){ll tax _“ __§‘$:ug§2) 1
y (~A+\)(\q 3)} (‘I4-|)(~‘+1)(Il
-\__ _\____¢_k; _ 't1.j~1\*:._F;A1\ ‘O
\_ _\_..§__ _5*?» 3* '°‘AL. - _.\....N £1 YA
Exercise 2.2: Page 24/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 24/29 - Available at www.MathCity.org
e\:_ =-—~-ch
Q1 .
O!‘ dx=-49. x‘+y"-.-.3axy‘= 0"
.15’
"u$"l‘I\.' - .. w~n¢\‘J-.'
I
--mu-»-r
in‘"-v‘MW' .*(
‘ -wi \f;.-iv§_I
3;
.5‘,~§
A=_,
_;,,v.p
,..‘.-,<».
,u ' ‘ ' -N,‘
w...-. >-_ ,1vyuu»-am... ‘ .~
>
Sol. Differehtiating w.r.t,. x, we get V.
d - J _
3x'+8y’ -‘:5-80 [ac 5% +.y.1]=6d
0; x'+y'g'x -a[x %+y.1]=0I
O'1
&+s~'H»*», G:'|' t‘I
(1 -1.y’-a x
50. y-cos (x+y) = Q
Sol. It canberewritten as
y = cos (x+y)
ghldu -A1-4°74‘ -mfg -.= '6'd‘) .3 Q3, *1
1
Differentiatnn jboth sides w.r.t. 'x, we gqt,
' d l-‘z'=—sin(.r+y) 1+‘—'Z-x ‘d [ 1
at - -*~<-~+»» - $**(*+=)-A”/Au
dx
#3; *$5-(\\+\>\-‘$3; 1 - Q;-\(\\-\-Q.
at \*‘5~'-(\-aw) . - $~:-h-\-‘=3.
Q sin (x + X) .
or (1x_ “ l+sin(X +7)SI. arctan (x+_\') =4 arcsin (4-” +x)
“4‘- I-ulna-"15"" 121-'2(1u-s) - $~.:'.‘( Z-\-1)
\_\-L_{*_»'(‘*".k¢) - --»»--_i._@T~,<=1’~a=:@~\>* “*3 \\\-(em)
._‘_._ + ..l__.. Ala. ,\+hu»>)‘ \+(“\,§~ A;
\__ .
‘ \-(_i’-+1.)‘ — ' WSL
in3'3'
+
__‘____..§ _ e" _____ 41 1 \
\*'u'*"’>t A‘ _!"(e*-13‘ 6* F-(€‘.-1>{ -. V\+(‘.\»'\$1K
"1 \ 1: \ _m..H‘*<*~»>’E311.» ="_'__"""‘ " “"""..".;.-_-I-'~ - ’ Ci
A* _ \'\'§‘\\'?'§t \-(e."-\-I3‘ g - ¢s1\l- \+l~A»u\\
>4@r_<;;‘_;, ~:
E
Exercise 2.2: Page 25/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 25/29 - Available at www.MathCity.org
5'2. x
-'7—[J\-(L’-+1)‘ -¢\°[!+Q-0-3)") [l+(lL+~s)‘- \_- 1 “"""“““"-"-——————-»~-- :_ ,_\ 1 “*““ M"-[ "(“"’5’ )J'T(¢“’+»\>‘ I*_—E’+ ] .
%1"('“”“*>‘-¢”(\*'(:*+»3")} 2 N b**‘°\1 - J \-(Z-WWX .
:1
:1-1)w L] 1-‘
Itqr
F"-1
of £3 I +<x+vi- xll-(<'*i+~>_’ ifI" \/I"-<¢'+x>’-¢-'<1+<x+.v>*>,
= a(t-sinl),y = a (1—cost)S01. Differentiating w.r.t. 1, wk: get
Nun
b
‘£5 = a (1-cost)
DJ
H5$153“
= a(-sin 1) = >a$:...'L
=.<iY 2!dt "dx
asinl\__ -._i.___._..__...-.
ijsm 5 cos 5
oz
bi“-
Zsin ""
55. x=acoa‘t,y=bsin"t_
a(1-cost) '
. t t
QO
U)
v\h)'\
‘C="__"'~‘-'COt';.sin
N"
Sk@L 71 I OKi:¥- §- 3 I'\D§£1t,
P-
8.2-gU
“‘\A)£,--Ié2—~#-Au
I
2
~ \»-m-t.- K“*3Q; ¢ a . '3Ca>£. (-g;_,¢)At 1-3aCJ>t.$;,,J¢
=\>s§’¢.u-A-4:-‘ll
‘O (3%i.£-cu‘)I '3kS$:\¥(;,¥
A-E 'd4's\.$.ZtC,
~ ~ a-aaaiea-Me.
_L_.‘~'-=._-£2 . -k_’q\..+,CL C;t_ CL
\+(1+~>)‘
Exercise 2.2: Page 26/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 26/29 - Available at www.MathCity.org
2.1
30! aa5‘? x ’ 1+‘! 0 y "' 1 + £2 _;'q_'s_;’-r“['!f"*fb
‘A; $1,; (- '
>¢ ’ ~ 'Sol. \>'~&&- ~»-~\-’<' ‘
§L . @.{¢~+~‘>-\-1-tw}** (\+x>1* .
= &»[Lb§kF](\-Hr‘-)1
Q1 _, $¢(\-V‘)dk‘. ,(\-Q-£1)‘
New él = $a{_(“\e§'1_'E'_.:E?_gf_'Q]C“ (\+§:,")1'
, 3q[1u19-v¢‘3(\+k?)" -
= 3Q Qt.)
(\+i,‘-3"‘
A’: (mi: at _(H_t_\)"
'“"‘ ‘éjf; = 'Ié"3I( ' 0+0->‘ '_s<~<\.u>
‘ ( 1.
55- y = (séc 1:3 + arcsec ac)‘
3 ..\ 1“
Sol. 5 = ($1:-x +s-L10“ n;.\g.u....k-1
~ £5)» = z(g¢,q-_’.,.g,1‘1§,§..($¢¢~;+¢¢r_x)
' ‘(M1 +<~1n.[sm.+.....;¢+ J.»-1, (st \ -\ 1 ( , *,§1L"-|.- '2. cx_~<L¢e.a){35¢_\_§;,__‘ _\_ \ \______ 5,5,
56. y = exP(arccsc 13”"\
4'-W-(‘I-A)Q
W.$-A-&-1h = ‘E“"*’_;1_._._ =1-6“ .L. Lu. 1‘
¢é¥0n) \’i“) \ oJ&{> \= =€.-_--
'1 !;*Zt‘E177 J v
Sol. ‘5 =
Exercise 2.2: Page 27/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 27/29 - Available at www.MathCity.org
A‘, Cvl;0:(\|~;) \
:7 ’ "1
=31..1
J2.A1
“P.*1‘
l 1 J \-‘iTML40-: f
57. _y_ 1 arcsin (ln_x) - In (arctan x)‘SJ: 5 ‘ -‘‘i\-_(-QM) — 3»~(i.?.‘.1)W. U-J-~\.£.‘_
S
\-(L.~¢)§ i - Rx"\ \
“F-(.3'-~\)‘ (\+v§§§1J.l.:a
58. y arc sin x - x arctan y = 1
S°1- ‘j$C.‘\x -1+cC.‘\\§ = \ ‘
\u~“5
' .\,._\._,\.t. -K
----*"+%'Z:1-Q2. 1._\.__._1\.1.,i\-RC’ J1 C M3‘ da _ 3 - = O
<52. (<2.-:31 _. _L~.> + 1.. 4&1.) ,0A1 H-91 J'§';;\ "
3*) (‘*‘;“"":}:l. = --:9-—--+'1"‘\;$4 ) . M,(S-s~3>") 53?
,3 .._..
-_ 1 .-. Y. " vii‘.
AL-((\*‘1'>S1E~_‘_};;__”L> . -u+§t;=\:>::»A‘ __....M
a ‘ ‘
= h‘::$~‘>) ' ‘qt;" IT? ““““"i"" ”—" "“*'
5_9. arcsin (In xy) = x-1 3/"‘K"*“")';’~’:‘ ‘ ‘*3' 1 ‘4.?;'
- i“(gMJT:U.~»\‘1§1 61' *3) = ‘*7-"3'é2'
d~A
—-L---_-.~_..=_, -.l-..-- \&2""\")~\ : \ '63.,1!-U»~\ss)" 7*‘) K 4' “tr, 1:‘ él
1% +3 * "*3JT;(_9»u~a)" S_\-\-1‘:-AJ
d4:_ ..---. ..___*5’;-113 ]\-(L‘»3‘-4: = *5,&\-(Lax-;)1 -3
‘Q.
‘L é; +3 = 13 _v'3T'_"(T'9.':;~:)-" + zx 3‘ ]\._(11,.,,¢,§:‘.§:’.
Exercise 2.2: Page 28/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 28/29 - Available at www.MathCity.org
x%( \-1§1tt:T;~3,) = 5(1y.:7_.@;,.,;¢_\) L_ 5x\(1;A
1*-1 - _.__( Y__:;,.r:>i:_‘l_ = ~>(*1\-1*» -1A1 X(\-1j1 1(\_1;
60* "1 '
)
. arcsec(.x +-_y)_ex = __1 ’ --- ~—--- ‘Z H
. x + y faS01. g";_\(‘_1+$) J ‘ '
. 1+: ’§\“- \M-~A.'\_- 1
uh») §(,1~,,.,§_, '
-1%;(f'+‘>§ -- = 9;; (1-Ma) .
______.‘__.____-_--- . 114-A3- .4? -= —-(\\%~".Sl- (\-\- é?-(§~5) §({.*_,)1_‘ ( J-A ) OH )
11+ 331 \ *\- A’_. Q = ______i._ag_--_.4- .-
(1Z‘+~:),! Li‘- \>)l—\ (X4-‘n)LA ' ‘K I. \~A+i- ¢(1~‘v3SR:‘a)‘-1 ...\ ._
-c»c».~,> .3 1»:-;.>>‘-1' _<'i;l§\7
(~44-»;)t(1.\\+ .. (14-“S~ 8 (~:»a\. !(,Q_§.,4\'-_; .= (- \- >(\\‘-\-‘::\§({+a)1—:-h.
1~1(1-*a§+ (H-5; -‘ii - (‘**'J1
35»
"1
' 1. \ T3- \ ; ’
(~:~~,>jTcI§§'-’\ = -(*~*>*J73*\¢-$%‘*~=>3?~7~=35
_ ‘_ i 1 x
(1+‘:)-A3; + %'_:(‘i:+u).§(~,{~+u§-\ = (‘A-P9)-8 (1"+\:)§({-$1)‘-\ -In-'>)§l\~‘>§'\ '7~‘*\‘**‘-‘\
ll; ‘\ ‘\.\ ~|_ ‘ '4-, ‘L 1 \. \. C(“H33 \.I-—"”‘ d .('l-i-‘))(1-\-‘9§X(_ 1- 3; x
u
(1~b>_‘ x-\-7}-I (14-'5)(1-\-5)-i -1'1 (_1-R1}
-—--~ xi ‘
P-"""" 1 * 3‘-\ -1x(_ -3
A» - _€(u,*'>§'(‘£‘-}'~>\>\(\%+‘@\}" "h*") (“'3' 1* >i ' 7 ___.- 77 i‘ ‘M-a“ (*A-\-‘))t-\- (X1-\-‘a\ J(_§“~~>(\1-\ A ' (
__ ._
Exercise 2.2: Page 29/29 - Available at www.MathCity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.2: Page 29/29 - Available at www.MathCity.org